If ` x = a+(1)/( a) and y = a - (1)/(a)` then value of ` x^(4) + y^(4) - 2 x ^(2) y^(2) is `

Options

A. 24
B. 18
C. 16
D. 12

Answer

To solve the problem, we need to find the value of \( x^4 + y^4 - 2x^2y^2 \) given that \( x = a + \frac{1}{a} \) and \( y = a - \frac{1}{a} \).

### Step 1: Calculate \( x^2 \) and \( y^2 \)

First, we calculate \( x^2 \):
\[ x^2 = \left(a + \frac{1}{a}\right)^2 = a^2 + 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \]

Next, we calculate \( y^2 \):
\[ y^2 = \left(a - \frac{1}{a}\right)^2 = a^2 - 2 \cdot a \cdot \frac{1}{a} + \left(\frac{1}{a}\right)^2 = a^2 - 2 + \frac{1}{a^2} \]

### Step 2: Substitute \( x^2 \) and \( y^2 \) into the expression

Now we substitute \( x^2 \) and \( y^2 \) into the expression \( x^4 + y^4 - 2x^2y^2 \).

Using the identity \( a^4 + b^4 - 2a^2b^2 = (a^2 - b^2)^2 \), we can rewrite our expression as:
\[ x^4 + y^4 - 2x^2y^2 = (x^2 - y^2)^2 \]

### Step 3: Calculate \( x^2 - y^2 \)

Now, we calculate \( x^2 - y^2 \):
\[ x^2 - y^2 = \left(a^2 + 2 + \frac{1}{a^2}\right) - \left(a^2 - 2 + \frac{1}{a^2}\right) \]
\[ = a^2 + 2 + \frac{1}{a^2} - a^2 + 2 - \frac{1}{a^2} = 4 \]

### Step 4: Calculate \( (x^2 - y^2)^2 \)

Now we can calculate \( (x^2 - y^2)^2 \):
\[ (x^2 - y^2)^2 = 4^2 = 16 \]

### Final Answer

Thus, the value of \( x^4 + y^4 - 2x^2y^2 \) is:
\[ \boxed{16} \]